Numerical investigation of two-dimensional fuzzy fractional heat problem with an external source variable

This study suggests a strategy for calculating the fuzzy analytical solutions to a two-dimensional fuzzy fractional-order heat problem including a diffusion variable connected externally. We propose Sawi residual power series scheme (SRPSS) which is the amalgamation of Sawi transform and residual power series scheme under the Caputo fractional differential operator. We demonstrate three different examples to derive the fuzzy fractional series solution which is characterized by its rapid convergence and easy finding of the unknown coefficients using the concept of limit at infinity. The most significant aspect of this scheme is that it derives the results without time effort compared with the traditional residual power series approach. Our findings confirm that the SRPSS is a robust and valuable method for approximating the solution of fuzzy fractional problems. Furthermore, we provide 2D and 3D symbolic representations to present the physical behavior of fuzzy fractional problems under the lower and upper bounded solutions.


Introduction
Many real-life events in physics, chemistry, control theory, and other scientific and engineering domains can be lucratively modeled using fractional derivatives [1][2][3].The most important reason is that fractional calculus allows for the appropriate modeling of real-world problems, which in turn depends on both the current and the preceding chronological period.The fractional differential equations (FDEs) have garnered attention from many scientific and engineering academics for their crucial role in comprehending various real-life processes in the natural sciences [4,5].These processes include mechanical systems, wave propagation phenomena, earthquake modeling, image processing, and control theory.Fractional calculus can be employed to characterize and reframe these processes as FDEs.The utilization of FDEs in the aforesaid and other phenomena is remarkable due to their characteristic of non-locality.In the same vein, the differential operators provide an excellent framework for describing the memory and genetic features of several processes and substances.
The theory of fuzzy sets was proposed by Zadeh [6] in 1965 as a natural progression from classical set theory.Therefore, many researchers have started to take an interest because of its proficiency in studying the unpredictability of facts and specifics.Fuzzy calculus has become highly important in a wide range of disciplines, including topology, fixed-point principle, integral inequalities, fractional algebra, bifurcations, image processing, recognition of patterns, intelligent systems, electronic devices, theory of control, and machine learning [7][8][9].Fuzzy set theory primarily enables us to generate new estimates and enhance the opportunities for efficiently managing and analyzing imprecise information.Several scholars expanded on this idea to construct primary fuzzy calculus based on fuzzy mapping and control.Fuzzy calculus involves the study of fuzzy sets and fuzzy numbers, which enable the representation and manipulation of uncertain and imprecise numbers.Thus, numerous scholars have studied FDEs [10][11][12].
Consider the fuzzy fractional heat problem in two dimensions, such as where α represents the Caputo fractional derivative and g 2 Cð½0; 1Þ � ½0; 1Þ � ½0; 1Þ; ½0; 1ÞÞ; f 2 ð½0; 1Þ � ½0; 1Þ; ½0; 1ÞÞ.The 2D heat problem is defined as the heat transfer occurring through an infinitely thin substance.In Eq (1), the parameter " W" shows the temperature of an object at any stage of the thin substance.This phenomenon of heat flow can be studied in numerous areas of physical science and technology.Hence, the examination of two-dimensional Fuzzy fractional heat equations holds significant relevance in diverse fields, including the analysis of heat transfer in substances with uncertain features, the numerical modeling of variation in temperature in biological structures, and the study of temperature phenomena in intricate systems with fuzzy variables.Fuzzy fractional differential and integral equations have received significant attention in the physical sciences during the past few decades.Many researchers have obtained the solution of fuzzy differential equations using various analytical and numerical techniques such as homotopy perturbation method [13], Fuzzy Laplace transform method [14], Crank-Nicolson method [15], Natural adomian decomposition method [16], SM's method [17].Akram and Bilal [18] utilized homotopy perturbation scheme for the analytical solution of fuzzy heat problem.Allahviranloo [19] applied a finite difference scheme to calculate the numerical solution of fuzzy heat and wave problems.Georgieva [20] utilized double Fuzzy Sumudu transform to solve partial Volterra fuzzy integro-differential equations.Mazandarani and Xiu [21] expanded the application of the differential transformation strategy to deal with fuzzy PDEs using strongly generalized differentiability.This method was discovered to be a direct and effective approach for getting analytical-numerical results.Osman et al. [22] introduced some analytical schemes for obtaining the solution of fuzzy fractional PDEs.
In the present study, we combine the Sawi transform and residual power series scheme to obtain the fuzzy results of fuzzy fractional heat problem in two-dimensional form.In the current framework, noteworthy findings are demonstrated without any restrictions being placed on the variables or assumptions being made regarding the parameters.The obvious benefit of this method is that it displays the outcomes in terms of power series, which enables one to get precise outcomes in a relatively short amount of time.Furthermore, we do not consider He's polynomials, which could potentially make the actual problems more complicated.Some graphical structures are presented for each problem, with different fractional order.The rest of paper is designed as follows: Section (2) consists of fuzzy definitions and the basic concept of Sawi transformation.We formulate the strategy of SRPSS in Sections (3) for fuzzy fractional problems.In Sections (4), we show the authenticity and reliability of SRPSS by considering three numerical applications in Section (4).The conclusion is summarized in Section (5).

Fuzzy integral and Sawi transform
Some concepts of fuzzy integrals and Sawi transform are presented in this section.
This formulation can be represented as a power series centered at τ = τ 0 , where τ is the variable and B represents the coefficients of the series.
Definition 2.7 Suppose W is a subset of the region δ and is a continuous fuzzy parameter defined on the interval [0, b].The Riemann-Liouville theory defines the fractional integral connected with δ as follows [31] GðZÞ dZ; ε 2 ð0; 1Þ: shows the is fuzzy continues and fuzzy Lebesgue integrable functions accordingly, thus the fuzzy fractional integral is stated as where such that the integral on the right side converges and m = dηe.Since η 2 (0, 1] so m = 1.Definition 2.9 The lower and upper boundaries of all fuzzy elements should meet these constraints.[32] • (i) The function f ðdÞ is non-reducing and constantly left on the time frame [0, 1].
• (ii) The function � f ðdÞ is a non-reducing and constantly right on the time frame [0, 1].

Construction of Sawi residual power series scheme (SRPSS)
This section introduces the Sawi residual power series scheme, which is utilized to approximate the solutions of the two-dimensional fuzzy fractional heat problem.This technique can be constructed directly without taking into account any assumptions.Let's assume the following nonlinear fuzzy fractional problem Step 1: By using ST to Eq (3), it gives By applying the definition of ST with constraints (4), we obtain or we may write it as Step 2: Applying inverse ST to Eq 5, we get with GðB; tÞ ¼ f ðB; mÞ þ S À 1 o a fg ðB; m; tÞg ½ �: Step 3: Let the general solution of Eq (3) can be written as: Thus, the truncated series results is Step 4: Suppose that ResW be residual function for Eq (8) is Then its truncated residual function yields as Now, the components of pn ðBÞ in Eq 8 can be obtained under the following condition:

Numerical applications
In this section, we provide the performance of SRPSS to calculate the fuzzy results of fuzzy heat problems in a two-dimensional form of fractional order.We consider three examples and derive the lower and upper bound solution at multiple fractional order of α.The results are shown in terms of fractional power series yielding the convergent solution just after a small number of repetitions.All the results are calculated with the help of Mathematica software.

Example 1
Suppose the following 2D fuzzy heat problem of fractional order as follows with following constraints WðB; m; 0Þ ¼ f e À ðBþmÞ : ð12Þ

Lower bound findings for 2D fuzzy problem.
The lower bound fuzzy problem can be derived from Eq (11) as follows with lower fuzzy initial constraints W 0 ðB; m; tÞ ¼ ðd À 1Þe À ðBþmÞ : ð14Þ Making the use of ST, we get Using the definition of ST in the sense of Caputo fractional order, we obtain By the use of inverse ST, it gives By employing the SRPSS, we obtain the following results: . . .: Thus, we can arrange it to the corresponding sequence Note: If g(B, μ, τ) = 0, then above equation becomes as This series turns to closed form solutions such as WðB; m; tÞ ¼ ðd À 1Þe À ðBþmÞ 4.1.2Upper bound findings for 2D fuzzy problem.The upper bound fuzzy problem can be derived from Eq (11) as follows with upper fuzzy initial constraints � W 0 ðB; m; tÞ ¼ ð1 À dÞe À ðBþmÞ : ð19Þ By employing the SRPSS, we obtain the following results: . . .: Therefore, we may organize it into the appropriate sequence Note: When g ðB; m; tÞ ¼ 0, the preceding equation is transformed into This series turns to closed form solutions such as We explain the physical nature of Example (4.1) in its several graphical structures.We include surface and contour plots that demonstrate the physical nature of the fuzzy heat fractional problem for lower bound approximation.

Lower bound findings for 2D fuzzy problem.
The lower bound fuzzy problem can be derived from Eq (23) as follows ð25Þ By employing the SRPSS, we obtain the following results: . . .: Therefore, we may organize it into the appropriate sequence Note: When g ðB; m; tÞ ¼ 0, the preceding equation is transformed as  This series turns to closed form solutions such as WðB; m; tÞ ¼ ðd À 1Þ sin ½pðB þ mÞ� 4.2.2Upper bound findings for 2D fuzzy problem.The upper bound fuzzy problem can be derived from Eq (23) as follows with upper fuzzy initial constraints By employing the SRPSS, we obtain the following results: . . .: Therefore, we may organize it into the appropriate sequence Note: When g ðB; m; tÞ ¼ 0, the preceding equation is transformed into This series turns to closed form solutions such as We explain the physical nature of Example (4.2) in its several graphical structures.We include surface and contour plots that demonstrate the physical nature of the fuzzy heat fractional problem for lower bound approximation.with lower fuzzy initial constraints By employing the SRPSS, one can compute the following results: . . .: Therefore, we may organize it into the appropriate sequence WðB; m; tÞ Note: When g ðB; m; tÞ ¼ 0, the preceding equation is transformed as This series turns to closed form solutions such as WðB; m; tÞ 4.3.2Upper bound findings for 2D fuzzy problem.The upper bound fuzzy problem can be derived from Eq (35) as follows with upper fuzzy initial constraints By employing the SRPSS, we obtain the following results: . . .: Therefore, we may organize it into the appropriate sequence Note: When g(B, μ, τ) = 0, the preceding equation is transformed into This series turns to closed form solutions such as We explain the physical nature of Example (4.3) in its several graphical structures.We include surface and contour plots that demonstrate the physical nature of the fuzzy heat fractional problem for for lower bound approximation.Our graphical structures of 3D and 2D show that the suggested scheme is valid and consistent for fuzzy fractional scenarios.

Conclusion
This paper introduces a novel scheme that combines the Sawi transform operator and the residual power series to address fuzzy heat time-fractional two-dimensional models with an external source variable.The advantage of this novel scheme is its capability to decrease the mathematical work required for determining the result in a power series structure whereas the unknown components are determined through a series of algebraic procedures.The suggested method is employed for analyzing three distinct physical models, and its validity has been demonstrated in effectively handling fractional nonlinear equations with both high precision and simplified calculation processes.The numerical findings indicate that the approximating results closely align with each other and have complete affirm with the precise results for α = 1.We examine the behavior of the derived results both numerically and graphically for a particular range of α values.One major benefit of SRPSS is its ability to derive series coefficients with few calculations, as it relies on the concept of limits instead of fractional derivatives.This indicates that the suggested scheme is suitable and extremely effective for generating approximate and analytical solutions to fuzzy fractional problems that occur in engineering and applied physics.We intend to expand the application of the Sawi residual power series scheme to address physical problems with more complex three-dimensional models in our upcoming work.
Figs 1 and 2 state the 3D graphical depictions of lower bound approximation at different coordinates of parameters at τ = 0.5 with −1 � B � 1 and 0 � μ � 1 for α = 0.75 and α = 1 respectively.Similarly, Figs 3 and 4 show the contour plots of lower bound approximation at different coordinates of parameters at τ = 0.5 with −2 � B � 2 and 0 � μ � 2 for α = 0.75 and α = 1 respectively.We include surface and contour plots that demonstrate the physical nature of the fuzzy heat fractional problem for upper bound approximation.Figs 5 and 6 state the 3D graphical depictions of upper bound approximation at different coordinates of parameters at τ = 0.5 with 0 � B � 1 and 0 � μ � 1 for α = 0.75 and α = 1 respectively.Similarly, Figs 7 and 8 show the contour plots of upper bound approximation at different coordinates of parameters at τ = 0.5 with 0 � B � 2 and 0 � μ � 2 for α = 0.75 and α = 1 respectively.In Figs 9 and 10, we demonstrate a 2D fuzzy structure for lower and upper bound approximation across the parameters δ = 0.2 and δ = 0.4.Our graphical structures of 3D and 2D show that the suggested scheme is valid and consistent for fuzzy fractional scenarios.